| L(s) = 1 | − 5.83i·2-s − 12.7i·3-s − 2.07·4-s − 74.5·6-s − 231. i·7-s − 174. i·8-s + 80.0·9-s − 223.·11-s + 26.5i·12-s + 169i·13-s − 1.35e3·14-s − 1.08e3·16-s − 1.32e3i·17-s − 467. i·18-s + 1.48e3·19-s + ⋯ |
| L(s) = 1 | − 1.03i·2-s − 0.818i·3-s − 0.0649·4-s − 0.845·6-s − 1.78i·7-s − 0.964i·8-s + 0.329·9-s − 0.556·11-s + 0.0531i·12-s + 0.277i·13-s − 1.84·14-s − 1.06·16-s − 1.11i·17-s − 0.339i·18-s + 0.940·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.173450762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173450762\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 + 5.83iT - 32T^{2} \) |
| 3 | \( 1 + 12.7iT - 243T^{2} \) |
| 7 | \( 1 + 231. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 223.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.32e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.32e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.10e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.44e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.64e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.69e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.25e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.56e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.07e5iT - 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.17993917496046678647235101385, −9.763426304902826856961786413279, −8.014167087783643786835093898706, −7.12822444823799503551366044239, −6.68880427713110027456291027379, −4.77003152287796400863329551455, −3.70814463369224347448169985725, −2.52127245616668615014073243234, −1.20294377828739099221631865920, −0.60258683422826490855403637497,
1.97651444843720018961651315817, 3.21712749742906457199113994717, 4.87355179432097574293405335919, 5.54697395640492290602952573348, 6.37309593066170997158860837479, 7.69408060288041963285126579369, 8.485965986655146697927157017750, 9.361469959881390226977711409513, 10.32587630083256019287308998931, 11.39770328462428093684051164946
